412 XV NORMED ALGEBRAS AND SPECTRAL THEORY 24. Let E be a separable Hilbert space and H a positive self-adjoint operator ; then / -f A# is invertible for all A > 0. For x e E and A > 0 let FA(x) = (A(/+ A/7)-1 'X\x). Show that F*(x) is increasing as a function of A, and that it is bounded if and only if x e H1/2(E). (Reduce to the case where H is a simple operator (15.11.3).) 25. Let E0 be a real Hilbert space and E the Hilbert space obtained by extending the field of scalars of E0 to C, so that every element of E is uniquely of the form x + iy with x, y e Eo , and (*' + iy' | x" + iy") = (*' | x") + (/ 1 /') + /(/ 1 x") - f(*' | y). Show that every self-adjoint operator ffQ on E0 extends uniquely to a self-adjoint operator H on E, having the same spectrum. 26. With the notation of Problem 2 of Section 13.13, suppose that for each compact subset K of X there exists a constant bk i> 0 such that (B) f \u\ JK for all us tf. This condition implies condition (A) of Problem 2, Section 13.13, but is not equivalent to (A). (a) In Problem 5(b) of Section 6,6, suppose that X is compact and that the functions fn are real-valued, bounded and measurable with respect to a positive measure ^ on X and that they satisfy the condition where ||/J = sup \fn(x)\. Show that the space & of functions which are /^-equivalent * e X to the functions belonging to the space denoted by E in the problem referred to, is such that H = tfFlJf is a Hilbert space isomorphic to E, and satisfying condition (B) above. (b) For every function /e &&(X, //,) with compact support K, there exists a function U7 E & such that (Uf \ u) = \ufd\ji for all u e tf. The class of Uf is uniquely deter- mined by the class of/, and we have (U'l <£ 6ic/2N2(/). Then the set ^ defined in Section 13.13, Problem 2(b) is also the closure of ^ in the set of the U/ for which / is ^0, compactly supported and belongs to && . Generalize the result of part (e) of this problem to the case where / € && is compactly supported and j>0 almost everywhere. Likewise, generalize part (f ) of the same problem. (c) Suppose that X is compact. Then Uf is defined for all functions / e J^jJ(X, ft), and we have N2(UO ^6xN2(/). If G -/is the class of U7, then G is a continuous positive self-adjoint operator on LR(X, /it). If F is the closure in LR of Gl/2(LJi) (which is the orthogonal supplement of Ker(<71/2) = Ker(C)), then the restriction of G112 to F is an isometry of the subspace F of LR onto the Hilbert space H (equipped with the norm |w|). Hence H = G1/2(L£). (d) Suppose that X is compact and that the "domination principle" is satisfied, in the form of (b): that is to say, if / e -5?R is ^0 almost everywhere, and if u e & is such that Uf(x) :g u(x) almost everywhere in the set of points x where f(x) > 0, then Uf(x) g u(x) almost everywhere in X. For each A > 0, put R* = G(I + AG)""1. If /e J£R is ^0 almost everywhere, and if g is a function whose class is equal toero on E0, is equal to Srl on EI, and is equal to Sn-j.Snl on £„ for all